Look and Say Polynomial

Just read 13112221 as “one 1, one 3, two 1s, three 2s, one 1”. This is look-and-say sequence
(A005150 in the OEIS).

Each next number is about 30% longer. More precisely, if denotes the number of digits of the -th member of the sequence, then

One can enumerate 92 possible subsequences, construct a transition matrix, interpret it as a
recurrence relation for the length of terms in the look-and-say sequence, recall that limiting
ratio of terms in the sequence is equal to the spectral radius of the transition matrix,
and finally find , the only positive rational root of Conway’s polynomial